Nonlinear Systems: Local Theory

Abstract

In Chapter 1 we saw that any linear system

$$ {\dot x}\; = \;A{\text{x}} $$

(1)

has a unique solution through each point x₀ in the phase space R _n ; the Solution is given by x(t) = e^Atx₀ and it is defined for all t ∈ R. In this chapter we begin our study of nonlinear systems of differential equations

$$ {\dot x}\; = \;{\text{f(x)}} $$

(2)

where f: E → Rⁿ and E is an open subset of Rⁿ. We show that under certain conditions on the function f, the nonlinear system (2) has a unique solution through each point x₀ ∈ E defined on a maximal interval of existence (α, β) ⊂ R. In general, it is not possible to solve the nonlinear system (2); however, a great deal of qualitative information about the local behavior of the solution is determined in this chapter. In particular, we establish the Hartman-Grobman Theorem and the Stable Manifold Theorem which show that topologically the local behavior of the nonlinear system (2) near an equilibrium point x₀ where f(x₀) = 0 is typically determined by the behavior of the linear system (1) near the origin when the matrix A = Df(x₀), the derivative of f at x₀. We also discuss some of the ramifications of these theorems for two-dimensional systems when det Df(x₀) ≠ 0 and cite some of the local results of Andronov et al. [A–I] for planar Systems (2) with det Df(x₀) = 0.